CAREER: Research and Training at the Intersection of Number Theory and Analysis

几千年来，素数一直是研究的自然对象，现在在现代数字通信加密系统中发挥着基础作用。尽管他们进行了长期的研究，但关于素数分布的许多问题仍未得到解决。质数也与类数的研究交织在一起，类数出现在许多数论设置中。类数的研究已经有200年了，尽管人们已经做出了精确的推测，但在很大程度上仍然是一个谜。Radon变换在数学谱上似乎很遥远，它量化了函数沿着低维曲面的“质量”分布；它们是计算机断层扫描医学成像理论的重要组成部分。Radon变换的研究是调和分析的中心领域，它与Carleson算子有着深远的联系，Carleson算子有助于回答关于傅里叶级数的历史性问题，并且与离散算术算子的新世界有着深远的联系，它将调和分析与数论相结合。这个项目是数论和谐波分析的交集，探索并连接了所有这些主题。在工作过程中，该项目将通过培养博士后、研究生暑期学校和儿童数学推广活动为数学界做出贡献。本研究的主要目标集中在数论与谐波分析交叉的五个项目上。首先，得到了关于任意次数域的类数可除性的新界。其次，将获得短字符和的新边界，它在历史上为l函数提供了重要的次凸性结果，现在在多维设置中有可能影响涉及丢芬图方程积分解计数的问题。第三，在丢番图方程领域，圆法的新变体和与之密切相关的多维振荡积分问题将得到发展。第四，将数论方法应用于谐波分析的设置，证明离散算子的新结果。最后，系统地研究了具有多项式相位和radon型行为的Carleson算子。

Prime numbers have been a natural object of study for thousands of years and now play a foundational role in modern encryption systems for digital communications. Despite their long study, many questions about the distribution of prime numbers remain unsolved. Primes are also intertwined with the study of class numbers, which appear in many number-theoretic settings. Class numbers have been studied for 200 years, but still remain largely mysterious, although precise conjectures have been developed. Seemingly far away on the mathematical spectrum, Radon transforms quantify the distribution of the "mass" of functions along lower-dimensional surfaces; they are a critical part of the theory underlying Computed Tomography medical imaging. The study of Radon transforms is a central area in harmonic analysis with far-reaching connections both to the Carleson operator, which was instrumental in answering a historic question on Fourier series, and to the new world of discrete arithmetic operators, which blends harmonic analysis with number theory. This project, which is at the intersection of number theory and harmonic analysis, explores and connects all of these themes. During the course of the work, the project will contribute to the mathematical community through training postdocs, a graduate summer school, and mathematical outreach activities for children.The major aims of this research center on five projects at the intersection of number theory and harmonic analysis. First, new bounds relating to the divisibility of class numbers of number fields of arbitrary degree will be obtained. Second, new bounds will be obtained for short character sums, which historically provided an important subconvexity result for L-functions, and now in a multi-dimensional setting have the potential to impact problems involving counting integral solutions to Diophantine equations. Third, in the realm of Diophantine equations, new variations on the circle method, and closely related questions on multi-dimensional oscillatory integrals, will be developed. Fourth, new results for discrete operators will be proved by adapting number-theoretic methods to the setting of harmonic analysis. Finally, a systematic investigation of Carleson operators with polynomial phases and Radon-type behavior will be carried out.